Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo (ABC) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers (UH) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes.


Introduction
The derivatives of non-integer orders, or fractional derivatives, are a mathematical concept that extends differentiation beyond integer orders.These types of derivatives have a large domain of applications in numerous fields, including physics, engineering, and finance, for theory.Non-singular fractional operators are important tools for researchers and practitioners seeking to understand and manipulate complex systems in a variety of contexts and they have stimulated a great deal of interest among researchers in their applicability to diverse problems, we indicate the readers to these works [10][11][12][13][14][15][16][17][18][19] and the references therein.Subsequently, authors of this work [20] popularized the AB definition to contain differentiation and integration with respect to non-negative, non-decreasing function, leading to the development of the G À AB operator.In 2023, Abdeljawad et al. [21] expanded this operator to higher-order fractional derivatives and integrals.Furthermore, this type of fractional derivative is a generalization of the traditional derivative, where the derivative is taken with respect to a function rather than a variable.This type of derivative is commonly used in fractional calculus with various operators to describe complex systems with non-integer order dynamics.In fact, by taking this fractional derivative, the researchers can better model the behavior of these systems and gain a deeper understanding of their underlying dynamics, see for example [22][23][24] and references cited therein.
An implicit differential equation involving a fractional derivative of an unknown function that appears implicitly in the equation has several benefits, including the ability to model complex systems with memory effects, non-local interactions, and anomalous diffusion.These equations also accurately describe physical phenomena such as transport in porous media or viscoelastic materials [25][26][27][28].In particular, Thabet and Kedim [29] studied a Hilfer fractional snap dynamic system on an infinite interval.Authors [30] discussed stability analysis of fractional pantograph implicit differential equations with initial boundary and impulsive conditions.Also, AB fractional derivative used to investigate the stability of implicit differential problem by authors [31].
In this situation, we would like to indicate that our contributions are interesting and the Eq (1.3) is new in the framework of G À ABC fractional order derivatives which include ABC derivative as a special case when GðuÞ ¼ u.Moreover, an approach analysis in this work is different about methods used in these works [32,33], and the Eq (1.3) covers many problems available in the literature studies, for instance, (i) the Eq (1. 3) reduces to problem (1.1) if μ ! 3, and the implicit term omitted; (ii) the Eq (1. 3) returns to problem (1.2) if we replace the operator ABC D m;G i by C D m i with omitting the implicit term.
The remaining parts of this paper are arranged as follows: Sec.2 is devoted to recalling the basic background materials related to fractional calculus and nonlinear analysis.Sec.3 discusses the existence and uniqueness theorems by using FPTs.Sec.4 is investigated UH stability.Finally, Sec.5 is dedicated to testing the effectiveness of main outcomes.

Preliminaries
In this situation, we present essential background material.Consider the space of continuous functions denoted by O ≔ CðJ ¼ ½i; r�; RÞ which is Banach space gifted with the norm kyk = sup υ2[ι,ρ] |y(υ)|.

Existence and uniqueness analysis
We introduce an equivalent integral fractional equation of the G À ABC IFDE (1.3).Regarding this, we first derive the following lemma: Lemma 3.
Proof At the beginning, we apply AB I m;G i on both sides of Eq (3.1) and using Lemma 2.4, we get So, due to the condition y(ι) = 0, we deduce that e 0 = 0. Thus, by substituting the value of e 0 and by taking the first derivative with respect to a function G, we find and due to the boundary condition ½yðiÞ� 0 G ¼ 0, we have e 1 = 0, which yields that Next, by applying the condition yðrÞ ¼ xyðsÞ; one has which implies that Hence, we deduce that Therefore, the proof is finished.As a consequence of the above lemma, we present the following essential result: For working analysis, we state the following conditions: (AS 1 ).There are the constants δ 1 , δ 2 , δ 3 > 0, such that for any y; ŷ 2 O and υ 2 J.
Proof We define a bounded ball D r as D r ¼ fy 2 O : kyk � rg.Regarding to show the continuity of @, let us taking the convergence sequence fy n g n2N to y in the ball ℧ B as n ! 1.Thus, by continuity of $ and by applying Lebesgue dominated convergence theorem, one has $ðu; y n ðuÞ; ABC D m;G i y n ðuÞÞ ¼ ð@yÞðuÞ: Hence, @ is continuous.Next, regarding to the growth condition, by applying (AS 1 ), we find jð@yÞðuÞj � jxjðGðuÞ À GðiÞÞ which implies that Therefore, in view of the Eqs (3.4) and (3.5), and taking supremum, one has k@yk � jxjðGðrÞ À GðiÞÞ Hence, k@yk � P 1 + P 2 kyk and this finishes the proof.
Proof According Theorem 3.5, @ is Lipschitz mapping and by Lemma 2.9, @ is χ-Lipschitz which implies that @ is χ-condensing.Now, due to Theorem 2.10, it remains to show that the set W is bounded, where W ¼ fy 2 O : y ¼ z@ðyÞ; for some z 2 ½0; 1�g: For end this, let y 2 W, therefore for each υ 2 J for some z 2 [0, 1], and by Theorem 3.3, we can derive that kyk ¼ kz @ðyÞk � P 1 þ P 2 kyk: Thus, kyk � P 1 1À P 2 ; which implies that W is a bounded set contained in O.In view of Theorem 2.10, implies that @ has at least one fixed point, which are act solutions of the G À ABC IFDE (1.3), and consequently W contains solutions of the Eq (1.3) is a bounded subset of O.

Conclusions
This manuscript dealt with a new class of G À ABC-IFDE (1.3) with higher orders belonging to the interval (2,3].The fundamental conditions of the existence and uniqueness of the solution for Eq (1.3) were established by Banach and topology degree theories.Moreover, the UH stability with its generalized was discussed.Finally, two application examples with illustrative  graphics and tables were provided to check the effectiveness of the main results with compare the main parameters.
The results of this study can be employed in new problems as special cases of the main Eq (1.3) by taking various functions of G. Furthermore, the GABC-IFDE (1.3) covers some problems are existing in the literature; for instance (i) the Eq (1.3) can be reduced to problem (1.1) if μ ! 3 and the implicit term omitted; (ii) the Eq (1.3) can be returned to problem (1.2) if we replace the operator ABC D m;G i by C D m i with omitting the implicit term.

Table 4 ,
we observe that P 3 � 1 for some values μ at function GðuÞ ¼ 0:9u 3 , thus for this reason and only at these values we can't say that the problem (5.2) has one solution.